Calculus Examples

$\int_{0}^{5} {(x - 2)}^{2} + 1 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{5} {(x - 2)}^{2} d x + \int_{0}^{5} d x$

Step 2

Let $u = x - 2$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = x - 2$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x - 2$ .

$\frac{d}{d x} [x - 2]$

Step 2.1.2

By the Sum Rule, the derivative of $x - 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- 2]$

Step 2.1.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

Step 2.2

Substitute the lower limit in for $x$ in $u = x - 2$ .

$u_{lower} = 0 - 2$

Step 2.3

Subtract $2$ from $0$ .

$u_{lower} = - 2$

Step 2.4

Substitute the upper limit in for $x$ in $u = x - 2$ .

$u_{upper} = 5 - 2$

Step 2.5

Subtract $2$ from $5$ .

$u_{upper} = 3$

Step 2.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = - 2$

$u_{upper} = 3$

Step 2.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{- 2}^{3} u^{2} d u + \int_{0}^{5} d x$

Step 3

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$\frac{1}{3} u^{3}]_{- 2}^{3} + \int_{0}^{5} d x$

Step 4

Apply the constant rule.

$\frac{1}{3} u^{3}]_{- 2}^{3} + x]_{0}^{5}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{1}{3} u^{3}$ at $3$ and at $- 2$ .

$(\frac{1}{3} \cdot 3^{3}) - \frac{1}{3} {(- 2)}^{3} + x]_{0}^{5}$

Step 5.2

Evaluate $x$ at $5$ and at $0$ .

$(\frac{1}{3} \cdot 3^{3}) - \frac{1}{3} {(- 2)}^{3} + (5) + 0$

Step 5.3

Simplify.

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Step 5.3.1

Raise $3$ to the power of $3$ .

$\frac{1}{3} \cdot 27 - \frac{1}{3} {(- 2)}^{3} + (5) + 0$

Step 5.3.2

Combine $\frac{1}{3}$ and $27$ .

$\frac{27}{3} - \frac{1}{3} {(- 2)}^{3} + (5) + 0$

Step 5.3.3

Cancel the common factor of $27$ and $3$ .

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Step 5.3.3.1

Factor $3$ out of $27$ .

$\frac{3 \cdot 9}{3} - \frac{1}{3} {(- 2)}^{3} + (5) + 0$

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 9}{3 (1)} - \frac{1}{3} {(- 2)}^{3} + (5) + 0$

Step 5.3.3.2.2

Cancel the common factor.

$\frac{3 \cdot 9}{3 \cdot 1} - \frac{1}{3} {(- 2)}^{3} + (5) + 0$

Step 5.3.3.2.3

Rewrite the expression.

$\frac{9}{1} - \frac{1}{3} {(- 2)}^{3} + (5) + 0$

Step 5.3.3.2.4

Divide $9$ by $1$ .

$9 - \frac{1}{3} {(- 2)}^{3} + (5) + 0$

Step 5.3.4

Raise $- 2$ to the power of $3$ .

$9 - \frac{1}{3} \cdot - 8 + (5) + 0$

Step 5.3.5

Multiply $- 8$ by $- 1$ .

$9 + 8 (\frac{1}{3}) + (5) + 0$

Step 5.3.6

Combine $8$ and $\frac{1}{3}$ .

$9 + \frac{8}{3} + (5) + 0$

Step 5.3.7

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$9 \cdot \frac{3}{3} + \frac{8}{3} + (5) + 0$

Step 5.3.8

Combine $9$ and $\frac{3}{3}$ .

$\frac{9 \cdot 3}{3} + \frac{8}{3} + (5) + 0$

Step 5.3.9

Combine the numerators over the common denominator.

$\frac{9 \cdot 3 + 8}{3} + (5) + 0$

Step 5.3.10

Simplify the numerator.

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Step 5.3.10.1

Multiply $9$ by $3$ .

$\frac{27 + 8}{3} + (5) + 0$

Step 5.3.10.2

Add $27$ and $8$ .

$\frac{35}{3} + (5) + 0$

Step 5.3.11

Add $5$ and $0$ .

$\frac{35}{3} + 5$

Step 5.3.12

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{35}{3} + 5 \cdot \frac{3}{3}$

Step 5.3.13

Combine $5$ and $\frac{3}{3}$ .

$\frac{35}{3} + \frac{5 \cdot 3}{3}$

Step 5.3.14

Combine the numerators over the common denominator.

$\frac{35 + 5 \cdot 3}{3}$

Step 5.3.15

Simplify the numerator.

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Step 5.3.15.1

Multiply $5$ by $3$ .

$\frac{35 + 15}{3}$

Step 5.3.15.2

Add $35$ and $15$ .

$\frac{50}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{50}{3}$

Decimal Form:

$16.\overline{6}$

Mixed Number Form:

$16 \frac{2}{3}$

Step 7